25 research outputs found
Loci of 3-periodics in an Elliptic Billiard: why so many ellipses?
We analyze the family of 3-periodic (triangular) trajectories in an Elliptic
Billiard. Specifically, the loci of their Triangle Centers such as the
Incenter, Barycenter, etc. Many points have ellipses as loci, but some are also
quartics, self-intersecting curves of higher degree, and even a stationary
point. Elegant proofs have surfaced for locus ellipticity of a few classic
centers, however these are based on laborious case-by-case analysis. Here we
present two rigorous methods to detect when any given Center produces an
elliptic locus: a first one which is a hybrid of numeric and computer algebra
techniques (good for fast detection only), and a second one based on the Theory
of Resultants, which computes the implicit two-variable polynomial whose zero
set contains the locus.Comment: 32 pages, 14 Figures, 6 Tables, 13 video
On the interplay between vortices and harmonic flows: Hodge decomposition of Euler's equations in 2d
Let be a compact manifold without boundary whose first homology is
nontrivial. Hodge decomposition of the incompressible Euler's equation in terms
of 1-forms yields a coupled PDE-ODE system. The -orthogonal components are
a `pure' vorticity flow and a potential flow (harmonic, with the dimension of
the homology). In this paper we focus on point vortices on a compact
Riemann surface without boundary of genus , with a metric chosen in the
conformal class.
The phase space has finite dimension . We compute a surface of
section for the motion of a single vortex () on a torus () with a
non-flat metric, that shows typical features of non-integrable 2-dof
Hamiltonians. In contradistinction, for flat tori the harmonic part is
constant. Next, we turn to hyperbolic surfaces (), having constant
curvature -1, with discrete symmetries. Fixed points of involutions yield
vortex crystals in the Poincar\'e disk. Finally we consider multiply connected
planar domains. The image method due to Green and Thomson is viewed in the
Schottky double. The Kirchhoff-Routh hamiltonian given in C.C. Lin's celebrated
theorem is recovered by Marsden-Weinstein reduction from to .
The relation between the electrostatic Green function and the hydrodynamical
Green function is clarified.
A number of questions are suggested
Can the Elliptic Billiard Still Surprise Us?
Can any secrets still be shed by that much studied, uniquely integrable,
Elliptic Billiard? Starting by examining the family of 3-periodic trajectories
and the loci of their Triangular Centers, one obtains a beautiful and
variegated gallery of curves: ellipses, quartics, sextics, circles, and even a
stationary point. Secondly, one notices this family conserves an intriguing
ratio: Inradius-to-Circumradius. In turn this implies three conservation
corollaries: (i) the sum of bounce angle cosines, (ii) the product of excentral
cosines, and (iii) the ratio of excentral-to-orbit areas. Monge's Orthoptic
Circle's close relation to 4-periodic Billiard trajectories is well-known. Its
geometry provided clues with which to generalize 3-periodic invariants to
trajectories of an arbitrary number of edges. This was quite unexpected.
Indeed, the Elliptic Billiard did surprise us!Comment: 19 pages, 16 figure
¿Flujo acústico, la “pequeña invención” de las cianobacterias?
Micro-engineering pumping devices without mechanical parts appeared “way back” in the early 1990’s. The working principle is acoustic streaming. Has Nature “rediscovered” this invention 2.7 Gyr ago? Strands of marine cyanobacteria Synechococcus swim 25 diameters per second without any visible means of propulsion. We show that nanoscale amplitude vibrations on the S-layer (a crystalline shell outside the outer membrane present in motile strands) and frequencies of the order of 0.5-1.5 MHz (achievable by molecular motors), could produce steady streaming slip velocities outside a (Stokes) boundary layer. Inside this boundary layer the flow pattern is rotational (hence biologically advantageous). In addition to this purported “swimming by singing”, we also indicate other possible instantiations of acoustic streaming. Sir James Lighthill has proposed that acoustic streaming occurs in the cochlear dynamics, and new findings on the outer hair cell membranes are suggestive. Other possibilities are membrane vibrations of yeast cells, enhancing its chemistry (beer and bread, keep it up, yeast!), squirming motion of red blood cells along capillaries, and fluid pumping by silicated diatoms.Los mecanismos de bombeo en microingeniería aparecieron al principio de la década de los 90. El principio detrás de esto es el de flujo acústico. ¿Ha descubierto la Naturaleza este invento de hace 2.700 millones de años? Algunas cianobacterias marinas de la especie Synechococcus nadan 25 diámetros por segundo sin ningún medio visible de propulsión. Especulamos en este artículo que vibraciones de amplitud de nanoescala del estrato S (una cáscara cristalina que cubre las membranas exteriores en las cepas móviles) y con frecuencias del orden de 0,5-1,5 MHz (y esto es factible por los motores moleculares), podrían producir velocidades de deslizamiento del fluido, en el exterior de la frontera de la región Stokes. Dentro de esta capa límite (que para nuestra sorpresa resulta ser relativamente ancha) el comportamiento del flujo es rotacional (y en consecuencia, ventajoso desde el punto de vista biológico). Adicionalmente a este supuesto mecanismo que se podria llamar “nadando cantando”, mostramos otros posibles ejemplos biológicos de corrientes acústicas. Sir James Lighthill ha sugerido que el flujo acústico también se da en la cóclea del oído de los mamíferos, y son muy sugerentes los nuevos hallazgos en las células ciliadas externas. Otras posibilidades son flujos acústicos producidos por vibraciones de las membranas en células de levadura, mejorando su química (¡cerveza y pan!), el contoneo de los glóbulos rojos en los tubos capilares y el bombeo de fluido producido por las diatomeas